Let \( a\in \mathbb {F}_{q}^{*} \) , and \( n\in \mathbb {N} \) . Based on the quadratic character of \( \mathbb {F}_{q}^{*} \) , two types of Jacobsthal sums [1]}, [2]} are defined by
\(\begin{split}I_{n}(a)&=\sum _{c\in \mathbb {F}_{q}^{*}}\eta (c^{n}+a),\\H_{n}(a)&=\sum _{c\in \mathbb {F}_{q}^{*}}\eta (c)\eta (c^{n}+a).\end{split}\) 
